How to validate website field in contact form 7?

Yes, you can add your own custom validation.
More info here.

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Here’s a working and tested example for your situation:

Use [text* your-website] inside your contact form 7 form.

Add this snippet to your theme’s functions.php, use a child-theme!

add_filter( 'wpcf7_validate_text*', 'custom_website_validation_filter', 20, 2 );
function custom_website_validation_filter( $result, $tag ) {
  if ( $tag->name == 'your-website' ) {

    $domain = isset( $_POST['your-website'] ) ? trim( $_POST['your-website'] ) : '';

    if ( ! checkdnsrr($domain, 'ANY') ) { // Check DNS records corresponding to a given Internet host name or IP address
      $result->invalidate( $tag, "We cannot find an active dns record for that website url?" );
    }
  }
  return $result;
}